Finite Group Databasebeta

← C2×C127×#1792 #44584286

Subgroups — C2×C127×#1792 #44584286

Order 32512.

4114 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×H(5,𝔽2)×C127C211yesmaximal, characteristic
2C22×#104×C127C212yesmaximal, complemented
3C2×#417×C127C212yesmaximal, complemented
4C22×#100×C127C211yesmaximal, characteristic, complemented
5C2×#390×C127C211yesmaximal, characteristic, complemented
6C127×#1792C218yesmaximal, direct product factor
7C2×H(5,𝔽2)×C127—28no—
8C2×H(5,𝔽2)×C127—24no—
9C2×H(5,𝔽2)×C127—24no—
10C2×H(5,𝔽2)×C127—22no—
11C2×H(5,𝔽2)×C127C418yescomplemented
12C2×H(5,𝔽2)×C127C2214yes—
13C22×#41×C127—22no—
14C22×#41×C127C2212yes—
15C23×D4×C127—22no—
16C23×D4×C127—22no—
17C23×D4×C127C2211yescharacteristic
18C22×M16×C127C2212yescomplemented
19C22×C4⋊C4×C127C2211yescharacteristic, complemented
20C22×C22⋊C4×C127—22no—
21C22×C42×C127C2211yescharacteristic, complemented
22C2×#104×C127—28no—
23C2×#104×C127C2218yescomplemented
24#417×C127—24no—
25#417×C127C2218yescomplemented
26C2×#103×C127—22no—
27C2×#102×C127—22no—
28C2×#100×C127—24no—
29C2×#100×C127C2214yescomplemented
30#390×C127—22no—
31#390×C127C2214yescomplemented
32C25×C127—22no—
33C25×C127D412yes—
34H(5,𝔽2)×C127—216no—
35H(5,𝔽2)×C127—232no—
36H(5,𝔽2)×C127—232no—
37H(5,𝔽2)×C127—216no—
38H(5,𝔽2)×C127—216no—
39H(5,𝔽2)×C127—28no—
40H(5,𝔽2)×C127C2×C4116yescomplemented
41C2×#41×C127—28no—
42C2×#41×C127—216no—
43C2×#41×C127—216no—
44C2×#41×C127—28no—
45C2×#41×C127—24no—
46C2×#41×C127—22no—
47C2×#41×C127—24no—
48C2×#41×C127—22no—
49C2×#41×C127C2×C418yes—
50C2×#41×C127D418yes—
51C2×#41×C127D414yes—
52C2×#41×C127C2314yes—
53C22×Q8×C127D412yes—
54C22×D4×C127—24no—
55C22×D4×C127—28no—
56C22×D4×C127—216no—
57C22×D4×C127—28no—
58C22×D4×C127—216no—
59C22×D4×C127—216no—
60C22×D4×C127—22no—
61C22×D4×C127—28no—
62C22×D4×C127—216no—
63C22×D4×C127—28no—
64C22×D4×C127—22no—
65C22×D4×C127—24no—
66C22×D4×C127—22no—
67C22×D4×C127C2×C414yes—
68C22×D4×C127D412yes—
69C22×D4×C127D414yes—
70C22×D4×C127D412yes—
71C22×D4×C127C2312yes—
72C23×C4×C127—22no—
73C23×C4×C127—22no—
74C23×C4×C127—21no—
75C23×C4×C127—21no—
76C23×C4×C127C2311yescharacteristic
77C2×M16×C127—22no—
78C2×M16×C127—24no—
79C2×M16×C127—24no—
80C2×M16×C127C2314yescomplemented
81C22×C8×C127—22no—
82C2×C4⋊C4×C127—22no—
83C2×C4⋊C4×C127—22no—
84C2×C4⋊C4×C127—21no—
85C2×C4⋊C4×C127C2312yescomplemented
86C2×C22⋊C4×C127—28no—
87C2×C22⋊C4×C127—48no—
88C2×C42×C127—24no—
89C2×C42×C127—21no—
90C2×C42×C127C2312yescomplemented
91#104×C127—216no—
92#103×C127—28no—
93#102×C127—28no—
94#100×C127—28no—
95C24×C127—28no—
96C24×C127—216no—
97C24×C127—22no—
98C24×C127—22no—
99C24×C127—22no—
100C24×C127—24no—
101C24×C127—44no—
102C24×C127—44no—
103C24×C127—84no—
104C24×C127—84no—
105C24×C127C22⋊C418yes—
106C24×C127C2×D411yescharacteristic
107C24×C127C2×D414yes—
108#41×C127—232no—
109#41×C127—264no—
110#41×C127—232no—
111#41×C127—28no—
112#41×C127—28no—
113#41×C127—28no—
114#41×C127—28no—
115#41×C127—28no—
116#41×C127—24no—
117#41×C127—24no—
118#41×C127C22⋊C4116yescomplemented
119#41×C127C22×C418yes—
120#41×C127C2×D418yes—
121C2×Q8×C127—22no—
122C2×Q8×C127—24no—
123C2×Q8×C127C22⋊C418yescomplemented
124C2×Q8×C127C2×D414yes—
125C2×D4×C127—216no—
126C2×D4×C127—232no—
127C2×D4×C127—232no—
128C2×D4×C127—264no—
129C2×D4×C127—232no—
130C2×D4×C127—264no—
131C2×D4×C127—28no—
132C2×D4×C127—232no—
133C2×D4×C127—24no—
134C2×D4×C127—28no—
135C2×D4×C127—24no—
136C2×D4×C127—232no—
137C2×D4×C127—28no—
138C2×D4×C127—28no—
139C2×D4×C127—28no—
140C2×D4×C127—24no—
141C2×D4×C127—24no—
142C2×D4×C127—22no—
143C2×D4×C127—28no—
144C2×D4×C127—24no—
145C2×D4×C127—24no—
146C2×D4×C127—24no—
147C2×D4×C127C22⋊C418yes—
148C2×D4×C127C22⋊C418yescomplemented
149C2×D4×C127C22×C414yes—
150C2×D4×C127C2×D414yes—
151C2×D4×C127C2×D414yes—
152C22×C4×C127—28no—
153C22×C4×C127—216no—
154C22×C4×C127—22no—
155C22×C4×C127—28no—
156C22×C4×C127—24no—
157C22×C4×C127—28no—
158C22×C4×C127—24no—
159C22×C4×C127—24no—
160C22×C4×C127—22no—
161C22×C4×C127—22no—
162C22×C4×C127—42no—
163C22×C4×C127—42no—
164C22×C4×C127—44no—
165C22×C4×C127—41no—
166C22×C4×C127—41no—
167C22×C4×C127C22×C412yes—
168C22×C4×C127C2×D411yescharacteristic
169C22×C4×C127C2×D414yes—
170C22×C4×C127C2×D412yes—
171C22×C4×C127C2×D412yes—
172C22×C4×C127C2×D412yes—
173C22×C4×C127C2411yescharacteristic
174M16×C127—24no—
175M16×C127—24no—
176M16×C127—44no—
177C2×C8×C127—24no—
178C2×C8×C127—44no—
179C4⋊C4×C127—24no—
180C4⋊C4×C127—42no—
181C22⋊C4×C127—416no—
182C42×C127—24no—
183C42×C127—42no—
184C23×C127—22no—
185C23×C127—22no—
186C23×C127—232no—
187C23×C127—24no—
188C23×C127—24no—
189C23×C127—28no—
190C23×C127—28no—
191C23×C127—24no—
192C23×C127—28no—
193C23×C127—44no—
194C23×C127—416no—
195C23×C127—44no—
196C23×C127—44no—
197C23×C127—44no—
198C23×C127—44no—
199C23×C127—44no—
200C23×C127—88no—
201C23×C127—82no—
202C23×C127—82no—
203C23×C127—832no—
204C23×C127—88no—
205C23×C127—88no—
206C23×C127C2×C22⋊C412yes—
207C23×C127C2×C22⋊C418yes—
208C23×C127#12012yes—
209C23×C127#12011yescharacteristic
210C23×C127#12011yescharacteristic
211C23×C127C22×D411yescharacteristic
212Q8×C127—24no—
213Q8×C127—28no—
214Q8×C127C2×C22⋊C418yescomplemented
215D4×C127—264no—
216D4×C127—28no—
217D4×C127—28no—
218D4×C127—216no—
219D4×C127—28no—
220D4×C127—28no—
221D4×C127—24no—
222D4×C127—216no—
223D4×C127—28no—
224D4×C127C2×C22⋊C418yescomplemented
225C2×C4×C127—232no—
226C2×C4×C127—24no—
227C2×C4×C127—28no—
228C2×C4×C127—28no—
229C2×C4×C127—28no—
230C2×C4×C127—216no—
231C2×C4×C127—24no—
232C2×C4×C127—48no—
233C2×C4×C127—42no—
234C2×C4×C127—42no—
235C2×C4×C127—44no—
236C2×C4×C127—42no—
237C2×C4×C127—82no—
238C2×C4×C127—82no—
239C2×C4×C127C2×C22⋊C414yes—
240C2×C4×C127C2×C22⋊C414yes—
241C2×C4×C127C2×C22⋊C412yes—
242C2×C4×C127#12014yes—
243C2×C4×C127#12014yes—
244C2×C4×C127#12012yes—
245C2×C4×C127#12012yes—
246C2×C4×C127C23×C411yescharacteristic
247C2×C4×C127C22×D412yes—
248C2×C4×C127C22×D412yes—
249C2×C4×C127C22×D411yescharacteristic
250C8×C127—44no—
251C22×C127—24no—
252C22×C127—24no—
253C22×C127—21no—
254C22×C127—21no—
255C22×C127—21no—
256C22×C127—21no—
257C22×C127—28no—
258C22×C127—28no—
259C22×C127—48no—
260C22×C127—416no—
261C22×C127—416no—
262C22×C127—44no—
263C22×C127—44no—
264C22×C127—44no—
265C22×C127—44no—
266C22×C127—84no—
267C22×C127—88no—
268C22×C127—832no—
269C22×C127#37912yes—
270C22×C127#37912yes—
271C22×C127#45312yes—
272C22×C127#45512yes—
273C22×C127C22×C22⋊C411yescharacteristic
274C22×C127C2×#12011yescharacteristic
275C22×C127C2×#12011yescharacteristic
276C4×C127—28no—
277C4×C127—44no—
278C4×C127—82no—
279C4×C127#37914yes—
280C4×C127C22×C22⋊C412yes—
281C4×C127C2×#12012yes—
282C2×#1792C12711yesmaximal, characteristic, direct product factor, Sylow
283C2×C127—21no—
284C2×C127—21no—
285C2×C127—22no—
286C2×C127—48no—
287C2×C127—48no—
288C2×C127#179214yesdirect product factor
289C2×C127C2×#37911yescharacteristic
290C2×C127C2×#45311yescharacteristic
291C2×C127C2×#45511yescharacteristic
292C22×H(5,𝔽2)C2×C12711yescharacteristic
293C22×#104C2×C12712yescomplemented
294C2×#417C2×C12712yescomplemented
295C22×#100C2×C12711yescharacteristic, complemented
296C2×#390C2×C12711yescharacteristic, complemented
297#1792C2×C12718yesdirect product factor
298C127C2×#179211yescharacteristic, direct product factor, Sylow
299C2×H(5,𝔽2)—28no—
300C2×H(5,𝔽2)—24no—
301C2×H(5,𝔽2)—24no—
302C2×H(5,𝔽2)—22no—
303C2×H(5,𝔽2)C4×C12718yescomplemented
304C2×H(5,𝔽2)C22×C12714yes—
305C22×#41—22no—
306C22×#41C22×C12712yes—
307C23×D4—22no—
308C23×D4—22no—
309C23×D4C22×C12711yescharacteristic
310C22×M16C22×C12712yescomplemented
311C22×C4⋊C4C22×C12711yescharacteristic, complemented
312C22×C22⋊C4—22no—
313C22×C42C22×C12711yescharacteristic, complemented
314C2×#104—28no—
315C2×#104C22×C12718yescomplemented
316#417—24no—
317#417C22×C12718yescomplemented
318C2×#103—22no—
319C2×#102—22no—
320C2×#100—24no—
321C2×#100C22×C12714yescomplemented
322#390—22no—
323#390C22×C12714yescomplemented
324C25—22no—
325C25D4×C12712yes—
326H(5,𝔽2)—216no—
327H(5,𝔽2)—232no—
328H(5,𝔽2)—232no—
329H(5,𝔽2)—216no—
330H(5,𝔽2)—216no—
331H(5,𝔽2)—28no—
332H(5,𝔽2)C2×C4×C127116yescomplemented
333C2×#41—28no—
334C2×#41—216no—
335C2×#41—216no—
336C2×#41—28no—
337C2×#41—24no—
338C2×#41—22no—
339C2×#41—24no—
340C2×#41—22no—
341C2×#41C2×C4×C12718yes—
342C2×#41D4×C12718yes—
343C2×#41D4×C12714yes—
344C2×#41C23×C12714yes—
345C22×Q8D4×C12712yes—
346C22×D4—24no—
347C22×D4—28no—
348C22×D4—216no—
349C22×D4—28no—
350C22×D4—216no—
351C22×D4—216no—
352C22×D4—22no—
353C22×D4—28no—
354C22×D4—216no—
355C22×D4—28no—
356C22×D4—22no—
357C22×D4—24no—
358C22×D4—22no—
359C22×D4C2×C4×C12714yes—
360C22×D4D4×C12712yes—
361C22×D4D4×C12714yes—
362C22×D4D4×C12712yes—
363C22×D4C23×C12712yes—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4—21no—
367C23×C4—21no—
368C23×C4C23×C12711yescharacteristic
369C2×M16—22no—
370C2×M16—24no—
371C2×M16—24no—
372C2×M16C23×C12714yescomplemented
373C22×C8—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4C23×C12712yescomplemented
378C2×C22⋊C4—28no—
379C2×C22⋊C4—48no—
380C2×C42—24no—
381C2×C42—21no—
382C2×C42C23×C12712yescomplemented
383#104—216no—
384#103—28no—
385#102—28no—
386#100—28no—
387C24—28no—
388C24—216no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—24no—
393C24—44no—
394C24—44no—
395C24—84no—
396C24—84no—
397C24C22⋊C4×C12718yes—
398C24C2×D4×C12711yescharacteristic
399C24C2×D4×C12714yes—
400#41—232no—
401#41—264no—
402#41—232no—
403#41—28no—
404#41—28no—
405#41—28no—
406#41—28no—
407#41—28no—
408#41—24no—
409#41—24no—
410#41C22⋊C4×C127116yescomplemented
411#41C22×C4×C12718yes—
412#41C2×D4×C12718yes—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8C22⋊C4×C12718yescomplemented
416C2×Q8C2×D4×C12714yes—
417C2×D4—216no—
418C2×D4—232no—
419C2×D4—232no—
420C2×D4—264no—
421C2×D4—232no—
422C2×D4—264no—
423C2×D4—28no—
424C2×D4—232no—
425C2×D4—24no—
426C2×D4—28no—
427C2×D4—24no—
428C2×D4—232no—
429C2×D4—28no—
430C2×D4—28no—
431C2×D4—28no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—28no—
436C2×D4—24no—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4C22⋊C4×C12718yes—
440C2×D4C22⋊C4×C12718yescomplemented
441C2×D4C22×C4×C12714yes—
442C2×D4C2×D4×C12714yes—
443C2×D4C2×D4×C12714yes—
444C22×C4—28no—
445C22×C4—216no—
446C22×C4—22no—
447C22×C4—28no—
448C22×C4—24no—
449C22×C4—28no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4C22×C4×C12712yes—
460C22×C4C2×D4×C12711yescharacteristic
461C22×C4C2×D4×C12714yes—
462C22×C4C2×D4×C12712yes—
463C22×C4C2×D4×C12712yes—
464C22×C4C2×D4×C12712yes—
465C22×C4C24×C12711yescharacteristic
466M16—24no—
467M16—24no—
468M16—44no—
469C2×C8—24no—
470C2×C8—44no—
471C4⋊C4—24no—
472C4⋊C4—42no—
473C22⋊C4—416no—
474C42—24no—
475C42—42no—
476C23—22no—
477C23—22no—
478C23—232no—
479C23—24no—
480C23—24no—
481C23—28no—
482C23—28no—
483C23—24no—
484C23—28no—
485C23—44no—
486C23—416no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—82no—
493C23—82no—
494C23—88no—
495C23—88no—
496C23—88no—
497C23—832no—
498C23C2×C22⋊C4×C12712yes—
499C23C2×C22⋊C4×C12718yes—
500C23#120×C12711yescharacteristic
501C23#120×C12712yes—
502C23#120×C12711yescharacteristic
503C23C22×D4×C12711yescharacteristic
504Q8—24no—
505Q8—28no—
506Q8C2×C22⋊C4×C12718yescomplemented
507D4—264no—
508D4—28no—
509D4—28no—
510D4—216no—
511D4—28no—
512D4—28no—
513D4—24no—
514D4—216no—
515D4—28no—
516D4C2×C22⋊C4×C12718yescomplemented
517C2×C4—232no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—216no—
523C2×C4—24no—
524C2×C4—42no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4C2×C22⋊C4×C12714yes—
532C2×C4C2×C22⋊C4×C12714yes—
533C2×C4C2×C22⋊C4×C12712yes—
534C2×C4#120×C12714yes—
535C2×C4#120×C12714yes—
536C2×C4#120×C12712yes—
537C2×C4#120×C12712yes—
538C2×C4C23×C4×C12711yescharacteristic
539C2×C4C22×D4×C12712yes—
540C2×C4C22×D4×C12712yes—
541C2×C4C22×D4×C12711yescharacteristic
542C8—44no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—24no—
548C22—24no—
549C22—28no—
550C22—28no—
551C22—48no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—416no—
557C22—416no—
558C22—84no—
559C22—88no—
560C22—832no—
561C22#379×C12712yes—
562C22#379×C12712yes—
563C22#453×C12712yes—
564C22#455×C12712yes—
565C22C22×C22⋊C4×C12711yescharacteristic
566C22C2×#120×C12711yescharacteristic
567C22C2×#120×C12711yescharacteristic
568C4—28no—
569C4—44no—
570C4—82no—
571C4#379×C12714yes—
572C4C22×C22⋊C4×C12712yes—
573C4C2×#120×C12712yes—
574C2—21no—
575C2—21no—
576C2—22no—
577C2—48no—
578C2—48no—
579C2C127×#179214yesdirect product factor
580C2C2×#379×C12711yescharacteristic
581C2C2×#453×C12711yescharacteristic
582C2C2×#455×C12711yescharacteristic
subgroupG/H
center, socleC23×C127#120
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC22×H(5,𝔽2)×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C23×C4×C127 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×#104×C127C2semidirect
3. C2×#417×C127C2semidirect
4. C22×#100×C127C2semidirect
5. C2×#390×C127C2semidirect
6. C127×#1792C2direct (N × H)
11. C2×H(5,𝔽2)×C127C4semidirect
18. C22×M16×C127C22semidirect
19. C22×C4⋊C4×C127C22semidirect
21. C22×C42×C127C22semidirect
23. C2×#104×C127C22semidirect
25. #417×C127C22semidirect
29. C2×#100×C127C22semidirect
31. #390×C127C22semidirect
40. H(5,𝔽2)×C127C2×C4semidirect
80. C2×M16×C127C23semidirect
85. C2×C4⋊C4×C127C23semidirect
90. C2×C42×C127C23semidirect
118. #41×C127C22⋊C4semidirect
123. C2×Q8×C127C22⋊C4semidirect
148. C2×D4×C127C22⋊C4semidirect
214. Q8×C127C2×C22⋊C4semidirect
224. D4×C127C2×C22⋊C4semidirect
282. C2×#1792C127direct (N × H)
288. C2×C127#1792direct (N × H)
293. C22×#104C2×C127semidirect
294. C2×#417C2×C127semidirect
295. C22×#100C2×C127semidirect
296. C2×#390C2×C127semidirect
297. #1792C2×C127direct (N × H)
298. C127C2×#1792direct (N × H)
303. C2×H(5,𝔽2)C4×C127semidirect
310. C22×M16C22×C127semidirect
311. C22×C4⋊C4C22×C127semidirect
313. C22×C42C22×C127semidirect
315. C2×#104C22×C127semidirect
317. #417C22×C127semidirect
321. C2×#100C22×C127semidirect
323. #390C22×C127semidirect
332. H(5,𝔽2)C2×C4×C127semidirect
372. C2×M16C23×C127semidirect
377. C2×C4⋊C4C23×C127semidirect
382. C2×C42C23×C127semidirect
410. #41C22⋊C4×C127semidirect
415. C2×Q8C22⋊C4×C127semidirect
440. C2×D4C22⋊C4×C127semidirect
506. Q8C2×C22⋊C4×C127semidirect
516. D4C2×C22⋊C4×C127semidirect
579. C2C127×#1792direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×H(5,𝔽2)×C1272
C22×#104×C1272
C2×#417×C1272
C22×#100×C1272
C2×#390×C1272
C127×#17922
C2×#1792127